Explicit description of generic representations for quivers of type \(\mathbb{A}_n\) or \(\mathbb{D}_n\) (Q5964512)

Let \(Q\) be a Dynkin quiver. Let rep\(Q\) denote the category of finite dimensional representations of \(Q\) over an algebraically closed field \(k\). For a dimension vector \(\mathbf{d}=(d_1,\cdots,d_n)\in \mathbb N^n\); the affine space rep\((Q,\mathbf{d})\) of representations \(X\) of \(Q\) over \(k\) with \(X(i)=k^{d_i}\) for any vertex \(i\) of \(Q\) admits an action of the affine algebraic group \(\mathrm{Gl}(\mathbf{d})=\prod_{i=1}^{i=n}Gl(d_i).\) Orbits in rep\((Q,\mathbf{d})\) are just isomorphism classes of representations in rep\(Q\). As \(Q\) is of finite representation type, rep\((Q,\mathbf{d})\) contains finitely many orbits and thus a unique dense orbit with respect to the Zariski topology. A representation in the dense orbit is called generic. A representation \(T\) is generic if and only if Ext\(^1_Q(T,T)=0\), see [\textit{C. M. Ringel}, Invent. Math. 58, 217--239 (1980; Zbl 0433.15009)]. In the paper under review, the author gives an explicit description of a generic representation of dimension vector \(\mathbf{d}\) for Dynkin quivers of type \(A_n\) and \(D_n\).